2013 Issue 431

  1. Which number is greater \[A=\left(1+\frac{1}{2013}\right)\left(1+\frac{1}{2013^{2}}\right)\ldots\left(1+\frac{1}{2013^{n}}\right)\] where $n$ is a positive integer, or ${\displaystyle B=\frac{2013^{2}-1}{2012^{2}-1}}$?.
  2. Given four points in the plane such that no pair of points has distance less than $\sqrt{2}$ cm. Prove that there exists two of them having a distance greater than or equal to $2$ cm. 
  3. Find the last two digits of the number  \[2003^{2004^{\mathstrut^{.^{.^{.^{2013}}}}}}.\]
  4. Find the maximum and minimum value of the expression \[P=27\sqrt{x}+8\sqrt{y}\] where $x,y$ are non-negative real numbers satisfying \[x\sqrt{1-y^{2}}+y\sqrt{2-x^{2}}=x^{2}+y^{2}.\]
  5. Let $ABCD$ be a cyclic quadrilateral, inscribed in circle $(O)$. $I$ and $J$ are the midpoints of $BD$ and $AC$ respectively. Prove that $BD$ is the angle bisector of angle $AIC$ if and only if $AC$ is the angle bisector of angle $BJD$. 
  6. Solve the following system of equations \[\begin{cases} x^{3}(1-x)+y^{3}(1-y) & =12xy+18\\ |3x-2y+10|+|2x-3y| & =10 \end{cases}.\]
  7. Determine the greatest value of the expression \[E=a^{2013}+b^{2013}+c^{2013},\] where $a,b,c$ are real numbers satisfying \[a+b+c=0,\quad a^{2}+b^{2}+c^{2}=1.\]
  8. Let $S.ABC$ be a triangular pyramid, $G$ is the centroid of the base triangle $ABC$, $O$ is the midpoint of $SG$. A moving plane $(\alpha)$ through $O$ meets the edges $SA$, $SB$, $SC$ at $A',B'C'$ respectively. Prove that \[\frac{SA'^{2}}{AA'^{2}}+\frac{SAB'}{BB'^{2}}+\frac{SC'^{2}}{CC'^{2}}\geq\frac{AA'^{2}}{SA'^{2}}+\frac{BB'^{2}}{SB'^{2}}+\frac{CC'^{2}}{SC'^{2}}.\]
  9. Find all natural numbers $n$ such that \[A=\left[\frac{n+3}{4}\right]+\left[\frac{n+5}{4}\right]+\left[\frac{n}{2}\right]+n^{2}+3n-1\] is a prime number, where $[x]$ denotes the greatest integer not exceeding $x$.
  10. Consider the real-valued function \[y=a\sin(x+2013)+\cos2014x\] where $a$ is given real number. Let $M,N$ be the greatest and smallest values respectively of this function over $\mathbb{R}.$ Prove that $M^{2}+N^{2}\geq2$.
  11. Let $\{a_{n}\}$ be a sequence given by \[a_{1}=\frac{1}{2},\quad a_{n+1}=\frac{a_{n^{2}}}{a_{n^{2}-a_{n}+1}},\,n=1,2,\ldots\]
    a) Prove that the sequence $\{a_{n}\}$ converges to a finite limit and find this limit.
    b) Let $b_{n}=a_{1}+a_{2}+\ldots+a_{n}$ for each positive integer $n$. Determine the integer part $[b_{n}]$ and the limit $\lim_{n\to\infty}b_{n}$.
  12. Given four points $A,B,C,D$ on circle $(ABC)$ and $M$ is a point not on this circle. Let $T_{i}$ be the triangle whose three vertices are $3$ of $4$ given points, except point $i$ ($i=A,B,C,D$). Let $H_{i}$ be the triangle whose vertices are the feet of the perpendicular drawn from $M$ onto the edges (or extended edges) of triangles $T_{i}$ ($i=A,B,C,D$). Prove that
    a) The circumcenter of triangles $H_{i}$ ($i=A,B,C,D$) lie on the same circle, centered at $O'$.
    b) When $D$ moves on the circle $(ABC)$, $O'$ always lie on a fixed circle.




Mathematics & Youth: 2013 Issue 431
2013 Issue 431
Mathematics & Youth
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